Finite Group Databasebeta

← #10689348

Subgroups — #10689348

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74360C211yesmaximal, characteristic
2#70479C211yesmaximal, characteristic
3#70453C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic
5#69577C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#8882—31nomaximal, Sylow
9C2×#893—21no—
10C2×D3×C4⋊C4—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C2×#821—21no—
16C2×#818—21no—
17#4989—22no—
18#4987—21no—
19#4972—21no—
20C2×#807—22no—
21#4935C2211yescharacteristic
22C4×C4⋊Dic3C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4551—21no—
25#4551—21no—
26#4483C414yescomplemented
27D4×C4⋊C4—31no—
28#1947—31no—
29#1845—31no—
30#1802—31no—
31C2×D4⋊C8—31no—
32#1296D311yescharacteristic, complemented
33#1279—31no—
34C22×D12D411yescharacteristic
35C22×C4×D3—21no—
36C22×C4×D3—22no—
37C22×C3×C8—21no—
38C22×C3×C8—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—22no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54#893—42no—
55#893—42no—
56D3×C4⋊C4—42no—
57D3×C4⋊C4—42no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12C2×C412yes—
61C4×D12Q812yes—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8C2×C412yescomplemented
66C3⋊#95—22no—
67C3⋊#95—21no—
68#821—42no—
69#818—42no—
70#807—44no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8C2×C412yescomplemented
73C2×C4×D4—31no—
74C2×C4×D4—61no—
75C22×C4⋊C4—61no—
76C2×#107—61no—
77C2×#107—62no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×C4⋊C8C2×D311yescharacteristic
80C2×#102—61no—
81C2×C4×C8—31no—
82#389—61no—
83#385—62no—
84#384—31no—
85#384C2×D311yescharacteristic
86#380—61no—
87C4×C4⋊C4—31no—
88#340—61no—
89#340—61no—
90D4⋊C8—34no—
91C23×D3—21no—
92C22×C3×C4—22no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—41no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12C4⋊C412yes—
104C2×D12D812yes—
105C2×D12Q1612yes—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—41no—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—44no—
113C2×C4×D3—44no—
114C2×C4×D3—82no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×Q811yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—44no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—42no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3C4⋊C412yes—
143C4⋊Dic3D812yes—
144Dic3⋊C4—44no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C22×D4—31no—
151C23×C4—62no—
152C23×C4—61no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—61no—
156C4×D4—32no—
157C4×D4—32no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—122no—
161C4×D4—122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—122no—
172C2×C4⋊C4—122no—
173C2×C4⋊C4D1212yes—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—62no—
176C2×C42—61no—
177C2×C42C22×D311yescharacteristic
178#107—122no—
179#107—124no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8C4×D312yescomplemented
184#102—122no—
185C4×C8—32no—
186C4×C8—61no—
187#95—62no—
188#95—61no—
189C23×C3#12011yescharacteristic
190C22×D3—22no—
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C2×C4⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C22⋊Q811yescharacteristic
206C2×C3×C4C22⋊Q811yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4C2×D811yescharacteristic
209C2×C3×C4C2×D811yescharacteristic
210C2×C3×C4C2×Q1611yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228D12—22no—
229D12—22no—
230D12—22no—
231D12#10714yescomplemented
232C4×D3—42no—
233C4×D3—42no—
234C4×D3—42no—
235C4×D3—42no—
236C4×D3—88no—
237C3×C8—42no—
238C3×C8—81no—
239C3⋊C8—24no—
240C24—61no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C22×C4—22no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—124no—
254C22×C4—124no—
255C22×C4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4—242no—
261C22×C4C2×D1211yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—62no—
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—122no—
285C4⋊C4—124no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—124no—
293C42—121no—
294C42—121no—
295C42C2×C4×D311yescharacteristic
296C42#23511yescharacteristic
297C42#23711yescharacteristic
298C42D3×Q811yescharacteristic
299C22×C3#38511yescharacteristic
300C22×C3#44711yescharacteristic
301C22×C3#44711yescharacteristic
302C22×C3#45111yescharacteristic
303C22×C3#45211yescharacteristic
304C22×C3#45311yescharacteristic
305C22×C3#45611yescharacteristic
306C2×D3—24no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—42no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—82no—
316C3×C4C2×#10711yescharacteristic
317C3×C4#42811yescharacteristic
318C3×C4C4×D811yescharacteristic
319C3×C4#44011yescharacteristic
320C3×C4#47411yescharacteristic
321C3×C4#47811yescharacteristic
322C3×C4#48011yescharacteristic
323C3×C4#48411yescharacteristic
324Dic3—42no—
325Dic3—42no—
326Dic3—81no—
327Dic3—161no—
328C23—62no—
329C23—62no—
330C23—62no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23#88211yescharacteristic
336D4—34no—
337D4—62no—
338D4—62no—
339D4—62no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—248no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4C4×D1211yescharacteristic
363C2×C4#67⋊C411yescharacteristic
364C2×C4#88311yescharacteristic
365C2×C4D3×C4⋊C411yescharacteristic
366C2×C4#89611yescharacteristic
367C2×C4#89711yescharacteristic
368C2×C4D3×D811yescharacteristic
369C2×C4D3×D811yescharacteristic
370C2×C4#91411yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#91611yescharacteristic
374C2×C4D3×Q1611yescharacteristic
375C2×C4#91811yescharacteristic
376C8—42no—
377C8—64no—
378C8—81no—
379C2×C3#161911yescharacteristic
380C2×C3#163211yescharacteristic
381C2×C3#164711yescharacteristic
382C2×C3#164911yescharacteristic
383C2×C3#185211yescharacteristic
384C2×C3#185911yescharacteristic
385C2×C3#186211yescharacteristic
386D3—44no—
387C22—64no—
388C22—122no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22#466811yescharacteristic
395C22#474711yescharacteristic
396C22#475211yescharacteristic
397C22#477311yescharacteristic
398C22#477311yescharacteristic
399C22#480811yescharacteristic
400C22#480911yescharacteristic
401C4—21no—
402C4—21no—
403C4—82no—
404C4—122no—
405C4—122no—
406C4—241no—
407C4—481no—
408C4#475611yescharacteristic
409C4#477411yescharacteristic
410C4#481111yescharacteristic
411C4#481711yescharacteristic
412C4#486111yescharacteristic
413C4#487111yescharacteristic
414C4D3×#10711yescharacteristic
415C4#489011yescharacteristic
416C3#888211yescharacteristic, complemented, Sylow
417C2—124no—
418C2#6839011yescharacteristic
419C2#6843611yescharacteristic
420C2#6846111yescharacteristic
421C2#6849211yescharacteristic
422C2#6887411yescharacteristic
423C2#6889411yescharacteristic
424C2#6900211yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#8882
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #70453C2semidirect
5. #69577C2semidirect
6. C3×#1296C2semidirect
7. #67704C2semidirect
26. #4483C4semidirect
32. #1296D3semidirect
65. C3×C4⋊C8C2×C4semidirect
72. C4×C3⋊C8C2×C4semidirect
183. C4⋊C8C4×D3semidirect
231. D12#107semidirect
416. C3#8882semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#743602
#704792
#704532
C2×#44832
#695772
C3×#12962
#677042
#88823